Let $(\mathcal A,\mu)$ be an associative algebra. According to usual deformation theory, deformations of $(\mathcal A,\mu)$ as an associative algebra are controlled by a differential graded algebra (DGLA). The latter is known as the Hochschild DGLA $(H(\mathcal A),\delta_\mu,[\cdot,\cdot]_G)$ where:

-$H(\mathcal A)$ is the Hochschild graded vector space (where $H(\mathcal A)^i:=Hom(\mathcal A^{\otimes i+1},\mathcal A)$).

-$\delta_\mu$ is the Hochschild differential.

-$[\cdot,\cdot]_G$ is the Gerstenhaber bracket.

Note that the Hochschild operator is nilpotent and compatible with the Gerstenhaber bracket only if $\mu$ is associative.

Explicitly, a deformation $\star$ of $\mu\in H^{1}(\mathcal A)$ is a formal power series: $\star=\mu+\epsilon x^{(1)}+\epsilon^2 x^{(2)}+\cdots$ where $x^{(i)}\in H^{1}(\mathcal A)$ such that $x:=\epsilon x^{(1)}+\epsilon^2 x^{(2)}+\cdots$ satisfies the Maurer-Cartan equation:

$$\delta_\mu x+\frac12[x,x]_G=0$$

thus ensuring the associativity of $\star$.

There is furthermore a natural notion of equivalence between deformations given by the following equivalence relation:

Two deformations $\star',\star$ of $\mu$ are said equivalent if there exists a formal power series:

$\phi:=Id+\epsilon\phi^{(1)}+\epsilon^2\phi^{(2)}$ with $\phi^{(i)}\in H^0(\mathcal A)$ such that:

$$\phi(f\star' g)=\phi(f)\star\phi(g) \text{ for all }f,g\in\mathcal A.$$

This notion of equivalence is also in some sense "controlled by" the Hochschild DGLA as the preceding relation can be expanded in powers of $\epsilon$ and formulated in terms of the Hochschild differential and the Gerstenhaber bracket. For example, the first order in $\epsilon$ reads:


This is useful for classifying inequivalent deformations using the cohomology of the Hochschild differential. For example, if two deformations of $\mu$ are equivalent up to order $r$ in $\epsilon$, the obstruction for them to be equivalent up to order $r+1$ lies in the first Hoschild cohomology group.

Let us now consider classifying deformations of a non-associative algebra $(V,\lambda)$. Naively, this problem seems trivial as there is no quadratic defining relation to inforce (as the vanishing of the associator in the associative case or of the Jacobiator in the Lie algebra case), so that any formal power series

$\star=\lambda+\epsilon x^{(1)}+\epsilon^2 x^{(2)}+\cdots$

would qualify as a deformation of $\lambda$ (without any further relation on the $x$'s). In other words, since there is no quadratic relation, there is no DGLA controlling this deformation problem.

However, as in the associative case, there is a natural notion of equivalence relation between deformations given similarly by:

Two deformations $\star',\star$ of $\lambda$ are said equivalent if there exists a formal power series:

$\phi:=Id+\epsilon\phi^{(1)}+\epsilon^2\phi^{(2)}$ with $\phi^{(i)}\in H^0(V)$ such that:

$$\phi(f\star' g)=\phi(f)\star\phi(g) \text{ for all }f,g\in V.$$ As this notion of equivalence is the same as in the associative case, it must be in the same sense "controlled by" the Hochschild DGLA associated to $\lambda$. However, the non-associativity of $\lambda$ prevents $(H(V),\delta_\lambda,[\cdot,\cdot]_G)$ to be a DGLA.

Question:Is there a relevant algebraic structure for classifying inequivalent deformations of non-associative algebras?

  • 1
    $\begingroup$ The article of Glassman defines cohomology groups, which are relevant for classifying inequivalent deformations (like in the case of Lie algebras, done by Gerstenhaber in "Deformations of rings and algebras"). $\endgroup$ Sep 12, 2017 at 19:16
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    $\begingroup$ If you truncate Hochschild complex to leave only terms of degree 0 and 1, it will still be a dg-Lie algebra, and its deformation functor will be what you're looking for. The Maurer-Cartan equation will be vacuous, but the action of End(A) will be non-trivial. $\endgroup$ Apr 15, 2021 at 21:05


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